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A Representation Theorem for Archimedean Quadratic Modules on $*$-Rings

  Published:2009-03-01
 Printed: Mar 2009
  • Jakob Cimpri\v{c}
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Abstract

We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand--Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.
Keywords: Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometry Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometry
MSC Classifications: 16W80, 46L05, 46L89, 14P99 show english descriptions Topological and ordered rings and modules [See also 06F25, 13Jxx]
General theory of $C^*$-algebras
Other ``noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22]
None of the above, but in this section
16W80 - Topological and ordered rings and modules [See also 06F25, 13Jxx]
46L05 - General theory of $C^*$-algebras
46L89 - Other ``noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22]
14P99 - None of the above, but in this section
 

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